Anomalous quantized plateaus in two-dimensional electron gas with gate confinement

Quantum information can be coded by the topologically protected edges of fractional quantum Hall (FQH) states. Investigation on FQH edges in the hope of searching and utilizing non-Abelian statistics has been a focused challenge for years. Manipulating the edges, e.g. to bring edges close to each other or to separate edges spatially, is a common and essential step for such studies. The FQH edge structures in a confined region are typically presupposed to be the same as that in the open region in analysis of experimental results, but whether they remain unchanged with extra confinement is obscure. In this work, we present a series of unexpected plateaus in a confined single-layer two-dimensional electron gas (2DEG), which are quantized at anomalous fractions such as 9/4, 17/11, 16/13 and the reported 3/2. We explain all the plateaus by assuming surprisingly larger filling factors in the confined region. Our findings enrich the understanding of edge states in the confined region and in the applications of gate manipulation, which is crucial for the experiments with quantum point contact and interferometer.


II > I
We assume that regions I and II are both in FQH or IQH state. Their filling factors can be written as Ⅰ = + Ⅰ ′ and Ⅱ = + Ⅱ ′ , with as an integer and 0 < Ⅰ ′ < Ⅱ ′ ≤ 1. If transmitted edge currents obtain equilibration with reflected edge currents in region II, we can write down the equations of each "contact" in Supplementary Fig. 1a according to the Landauer-Buttiker formula 1-3 . Here, contacts labeled "U" and "L" are "virtual contacts", which indicate that edge modes inflow and outflow at these points are in equilibrium and share the same chemical potential.
And then can be derived as: which is the scenario we discuss in this work.

II < I
If II < I , edge currents will be partially reflected as they propagate from region I to region II, as shown in Supplementary Fig. 1b. In this case, equations of contacts are shown as follows: can be derived as: This is the common situation when measuring devices with lateral confinement, and it looks as if is measuring the Hall resistance in region II. However, this is correct only with the precondition II < I . Figure 1| Sketch of edge modes propagation and reflection when the filling factor in region II is larger (a) and smaller (b) than region I. a, When II > I , edge currents propagate in the same direction as that in Fig. 3a in the main text. Edge currents are reflected when propagating from region II towards region I. In region II, mixed edge currents get equilibrium before they reach the interface of different regions. Imaginary contacts labeled as U and L indicate that at these positions, edge modes are equilibrated, and the currents inflow and outflow share the same chemical potential. b, When II < I , edge currents are reflected when propagating from region I to region II. The reflected currents propagate along the interface on the side of region I.

Supplementary Note 2. Typical traces of , , , and
Supplementary Figure 2| Typical traces of , , , and versus magnetic field. L is the longitudinal resistance across the confined region and can be measured from contact 1 and 3 or from contact 2 and 4 ( Supplementary Fig. 1a). When D appears as anomalous plateaus, L appears as plateaus with finite values, rather than being zero. Source data are provided as a Source Data file.

Supplementary Note 3. Coexistence of plateaus and their relationship with electron density variation
In the main text, we attribute the appearance of anomalous plateaus to a gate-induced density increase in the confined region. In this section, the relationship between anomalous plateaus and the electron density variation in region II is discussed. Supplementary Fig. 3a shows D traces at three different gate conditions in 1 < ν < 2. Plateaus can appear together at the same gate voltage, such as the coexistence of K /(3/2), K /(10/7), K /(9/7) and K /(16/13) plateaus in the blue trace. This suggests that the emergence of these plateaus share the same origin, which can be explained by an electron density modulation in region II. To make it clear, the relationship between plateaus and II / I is illustrated in Supplementary Fig. 3b.
The y axis II / I represents the relative density between regions I and II. II / I should be larger than 1 in our experiments. The x axis is magnetic field, it corresponds to the filling factor of the open region IQH/FQH state. From the XY trace, the filling factor range for each IQH/FQH state can be obtained (defined by the plateau from XY ). As a simple estimation, we assume that the filling factor range for each state does not change when density varies. Then we know the filling factor range when region II will enter each IQH/FQH state, and we know I and II ranges when D will become plateaus. The filling factor range difference between the regions I and II determines the value of II / I at different magnetic fields. As consequence, the shapes in Supplementary Fig. 3b are drawn in relationship of II / I and B.
As Hall resistance of region II cannot be measured directly in our devices, we measure D instead. Anomalous plateaus can coexist at specific II / I values, as shown by the horizontal dashed lines in Supplementary Fig. 3b. And the three dashed lines correspond to the three D traces in identical colors in Supplementary Fig. 3a qualitatively.